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A210664 Square array read by upwards antidiagonals: T(m,n) is the number of simple 3-connected triangulations of a closed region in the plane with m+3 given external edges and 3n+m internal edges, m>=0, n>=1.

%I A210664 #23 Feb 25 2021 02:24:51
%S A210664 1,1,0,1,2,1,1,5,6,3,1,9,20,22,12,1,14,50,85,91,52,1,20,105,254,385,
%T A210664 408,241,1,27,196,644,1287,1836,1938,1173,1,35,336,1448,3696,6630,
%U A210664 9120,9614,5929,1,44,540,2967,9468,20790,34846,46805,49335,30880
%N A210664 Square array read by upwards antidiagonals: T(m,n) is the number of simple 3-connected triangulations of a closed region in the plane with m+3 given external edges and 3n+m internal edges, m>=0, n>=1.
%C A210664 A triangulation is simple if it contains no separating 3-cycle. There are n interior nodes and m+3 nodes on the boundary. - _Andrew Howroyd_, Feb 24 2021
%H A210664 Andrew Howroyd, <a href="/A210664/b210664.txt">Table of n, a(n) for n = 0..1325</a>
%H A210664 P. N. Rathie, <a href="http://dx.doi.org/10.1016/0095-8956(74)90055-0">A census of simple planar triangulations</a>, J. Combin. Theory, B 16 (1974), 134-138.
%H A210664 William T. Tutte, <a href="http://dx.doi.org/10.4153/CJM-1962-002-9">A census of planar triangulations</a>, Canad. J. Math. 14 (1962), 21-38.
%F A210664 From _Andrew Howroyd_, Feb 24 2021: (Start)
%F A210664 G.f. of row m > 0: R(x) satisfies g(x^2)^(m+1)*R(x*g(x^2)) = B(x^2) where g(x) is the g.f. of column 0 of A341856 and B(x) is the g.f. of column m of A341856.
%F A210664 G.f. of row m > 0: h(x)^(m+1)*B(x*h(x)^2) where 2-h(x) is the g.f. of A000256 and B(x) is the g.f. of column m of A341856.
%F A210664 (End)
%e A210664 Array begins:
%e A210664   1,  0,   1,    3,   12, ... (A000256)
%e A210664   1,  2,   6,   22,   91, ...
%e A210664   1,  5,  20,   85,  385, ...
%e A210664   1,  9,  50,  254, 1287, ...
%e A210664   1, 14, 105,  644, 3696, ...
%e A210664   1, 20, 196, 1448, 9468, ...
%e A210664   ...
%e A210664 From _Andrew Howroyd_, Feb 24 2021: (Start)
%e A210664 The array transposed for comparability with A341856 begins:
%e A210664 ==================================================
%e A210664 n\m |   0    1    2     3      4      5      6
%e A210664 ----+---------------------------------------------
%e A210664   1 |   1    1    1     1      1      1      1 ...
%e A210664   2 |   0    2    5     9     14     20     27 ...
%e A210664   3 |   1    6   20    50    105    196    336 ...
%e A210664   4 |   3   22   85   254    644   1448   2967 ...
%e A210664   5 |  12   91  385  1287   3696   9468  22131 ...
%e A210664   6 |  52  408 1836  6630  20790  58564 151146 ...
%e A210664   7 | 241 1938 9120 34846 116641 353056 983664 ...
%e A210664 (End)
%o A210664 (PARI) \\ here H is A000256 as g.f., U(n,m) is A341856 for m > 0.
%o A210664 H(n)={my(g=1+serreverse(x/(1+x)^4 + O(x*x^n) )); 2 - sqrt(serreverse(x*(2-g)^2*g^4)/x)}
%o A210664 U(n, m)={(3*(m+2)!*(m-1)!/(3*n+3*m+3)!)*sum(j=0, min(m, n-1), (4*n+3*m-j+1)!*(m+j+2)*(m-3*j)/(j!*(j+1)!*(m-j)!*(m-j+2)!*(n-j-1)!))}
%o A210664 R(N, m)={my(g=2-H(N)); Vec(if(m==0, 1-g, g^(m+1)*subst(O(x*x^N) + sum(n=1, N, U(n,m)*x^n), x, x*g^2)))}
%o A210664 M(m, n=m)={Mat(vectorv(m+1, i, R(n,i-1)))}
%o A210664 M(7) \\ _Andrew Howroyd_, Feb 23 2021
%Y A210664 Rows m=0..3 are A000256, A000139, A341920, A341921.
%Y A210664 Columns are A000012, A000096, A002415, A004305.
%Y A210664 Antidiagonal sums give A341922.
%Y A210664 Cf. A341856.
%K A210664 nonn,tabl
%O A210664 0,5
%A A210664 _N. J. A. Sloane_, Mar 28 2012
%E A210664 Terms a(21) and beyond from _Andrew Howroyd_, Feb 23 2021